Summary
- Gaussian processes (GPs) have two components, mean function and kernel function.
- Often the focus is put on the kernel function and the mean function assumed to be zero.
- Arbitrary choice of mean function can lead to unphysical results.
- Example: calculation of distance moduli of Type Ia supernovae under two cosmological models.
- Marginalise over a family of mean functions to remove the effect of mean function choice.
GP Notation
For data \(\{(\boldsymbol{x}_i, y_i); i = 1, \ldots, N\}\) with \(\boldsymbol{x}_i \in \mathbb{R}^d\) and \(y_i \in \mathbb{R}\),
\[y_i = f(\boldsymbol{x}_i) + \varepsilon_i\]
We propose a underlying function,
\[f(\cdot) \sim\mathcal{MVN} \left( \mu(\boldsymbol{x};\boldsymbol\theta_\mu), k(\boldsymbol{x}, \boldsymbol{x'}; \boldsymbol{\theta}_k) \right)\]
where \(\mu(\cdot)\) is the mean function and \(k(\cdot)\) is the covariance kernel function, with hyperparameters \(\boldsymbol\theta_\mu\) and \(\boldsymbol\theta_k\), respectively.
Mean Function
If we were to take many realisations of a GP, the mean of these over the support would be the specified mean function.
For example, \(\mu(\boldsymbol{x}) = \boldsymbol{0}\), \(k(x, x') = \exp\{-\|x - x'\|^2\}\):
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GP Regression
\[
\left[ {\begin{array}{c} \boldsymbol{y}\\ \boldsymbol{f^*}\\
\end{array} } \right] \sim \mathcal{N} \left(\left[ \begin{array}{c} \boldsymbol{\mu}\\ \boldsymbol{\mu^*} \\
\end{array} \right], \left[ \begin{array}{cc} k(\boldsymbol{x}, \boldsymbol{x}) + C& k(\boldsymbol{x}, \boldsymbol{x^*})\\ k(\boldsymbol{x^*}, \boldsymbol{x}) & k(\boldsymbol{x^*}, \boldsymbol{x^*}) \\\end{array}\right] \right)
\]
where \(\boldsymbol{x}\) are observed points whose values are \(\boldsymbol{y}\), and \(\boldsymbol{x^*}\) are target points with predicted values \(\boldsymbol{f^*}\).
The reconstruction \(\boldsymbol{f^*}\) is dependent on the choice of \(\boldsymbol{\mu^*}\) and \(k(\cdot)\) which themselves have hyperparameters \(\boldsymbol{\theta}\).
Conditional Distribution
We actually want the reconstruction conditioned on observations, \(\boldsymbol{y}\).
\[\boldsymbol{f^*} \vert \; \boldsymbol{y} \sim \mathcal{MVN}(\boldsymbol{\bar{f^*}}, \mathrm{Cov}(\boldsymbol{f^*}))\]
Chain Rule & Marginalisation
Joint probabilities expressed as conditional probabilities:
\[P(A, B) = P(A|B) \times P(B)\]
\[P(A,B | C) = P(A|B,C) \times P(B|C)\]
Variables can be “integrated out”:
\[\begin{align}
P(X) = &\int_y P(X, Y = y) \mathrm{d}y\\
= & \int_y P(X \vert Y = y) \times P(Y = y) \mathrm{d}y
\end{align}\]
GP Regression
Marginal likelihood of conditional distribution
\[p(\boldsymbol{y} | \boldsymbol{x}, \boldsymbol{\mu}, \boldsymbol{\theta}_k) = \int p(\boldsymbol{y}|\boldsymbol{f}, \boldsymbol{x}) \times p(\boldsymbol{f}|\boldsymbol{x}, \boldsymbol{\mu}, \boldsymbol{\theta}_k) \;d\boldsymbol{f}\]
We can choose values for \(\mu\) and \(\theta\) that maximise this quantity to get the “best fit” Gaussian process.
There are sometimes analytical solutions for this.
Data
- Pantheon SNIa compilation contains 1048 points upto \(z \simeq 2.3\).
- Mock data based on Pantheon with added Gaussian white noise.
- Two models for comparison
- Phenomenologically emergent dark energy (PEDE) model, and
- Chevellier-Polarski-Linder (CPL) parameterisation of DE equation of state.
Method
Apply GP regression with
- Different mean functions
- Different kernel functions
- Squared Exponential
- Matern 3/2
- Marginalise over different mean functions
Results: Zero mean function
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Results: Marginal Likelihood
\[k(x, x'; \sigma_f, \ell) = \sigma_f^2 \exp\left\{-\frac{1}{2}\left( \frac{x - x'}{\ell}\right)^2\right\}\qquad \eta, \ell \gt 0\]
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A Choice
Find the “best” hyperparameter settings by optimising the marginal likelihood. \[p(\boldsymbol{y} | \boldsymbol{x}, \boldsymbol{\mu}, \boldsymbol{\theta}_k) = \int p(\boldsymbol{y}|\boldsymbol{f}, \boldsymbol{x}) \times p(\boldsymbol{f}|\boldsymbol{x}, \boldsymbol{\mu}, \boldsymbol{\theta}_k) \;d\boldsymbol{f}\]
Marginalise over all values of hyperparameters to remove their effects. \[p(\boldsymbol{y} \vert \boldsymbol{x}, \boldsymbol{\mu}) = \iint p(\boldsymbol{y} \vert \boldsymbol{f}, \boldsymbol{x})\, p(\boldsymbol{f}\vert \boldsymbol{x}, \boldsymbol{\theta}) \mathrm{d}\boldsymbol{f}\mathrm{d}\boldsymbol\theta\]
Results: Full-marginalisation
\[p(\boldsymbol{y} \vert \boldsymbol{x}) = \iiint p(\boldsymbol{y} \vert \boldsymbol{f}, \boldsymbol{x})\, p(\boldsymbol{f}\vert \boldsymbol{x}, \boldsymbol{\mu}, \boldsymbol{\theta}) \mathrm{d}\boldsymbol{f}\mathrm{d}\boldsymbol\theta\mathrm{d}\boldsymbol\mu\]
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Different kernels
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Bias
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Conclusions of Paper
- Choice of mean function should not be arbitrary but carefully selected.
- Don’t use zero mean function for non-stationary quantities.
- Full marginalisation does not seem to be too sensitive to choice of family of mean functions.
- Difference between Squared Exponential and Matern kernels is negligible.
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